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Morphable model of quadrupeds skeletons for animating3D animals

Lionel Reveret, Laurent Favreau, Christine Depraz, Marie-Paule Cani

To cite this version:Lionel Reveret, Laurent Favreau, Christine Depraz, Marie-Paule Cani. Morphable model ofquadrupeds skeletons for animating 3D animals. ACM SIGGRAPH / Eurographics Symposium onComputer Animation, SCA’05, Jul 2005, Los Angeles, United States. �inria-00389338�

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Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2005)K. Anjyo, P. Faloutsos (Editors)

Morphable model of quadrupeds skeletonsfor animating 3D animals

Lionel Reveret, Laurent Favreau, Christine Depraz, Marie-Paule Cani

GRAVIR, INRIA

AbstractSkeletons are at the core of 3D character animation. The goal of this work is to design a morphable model of3D skeleton for four footed animals, controlled by a few intuitive parameters. This model enables the automaticgeneration of an animation skeleton, ready for character rigging, from a few simple measurements performed onthe mesh of the quadruped to animate.Quadruped animals - usually mammals - share similar anatomical structures, but only a skilled animator can eas-ily translate them into a simple skeleton convenient for animation. Our approach for constructing the morphablemodel thus builds on the statistical learning of reference skeletons designed by an expert animator. This raises theproblems of coping with data that includes both translations and rotations, and of avoiding the accumulation oferrors due to its hierarchical structure. Our solution relies on a quaternion representation for rotations and the useof a global frame for expressing the skeleton data. We then explore the dimensionality of the space of quadrupedskeletons, which yields the extraction of three intuitive parameters for the morphable model, easily measurableon any 3D mesh of a quadruped. We evaluate our method by comparing the predicted skeletons with user-definedones on one animal example that was not included into the learning database. We finally demonstrate the usabilityof the morphable skeleton model for animation.

Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics: Animation]:

1. Introduction

Skeleton construction and articulation placements are thefirst steps of character rigging. They involve the definitionand adjustment of numerous degrees of freedom, namely the3D position and orientation for each skeleton joint. Thesecomplex tasks are usually performed by a skilled animator.Tackling the problem in the case of virtual animals is evenmore complex than for virtual humans, since less anatomicaldata is available.

This paper shows that statistical analysis can be applied ona small set of skeleton models built by an expert animator togenerate a morphable model of quadrupeds skeletons, easilyadaptable to a wide variety of animals.

In the trade-off between fully procedural methods ver-sus data-oriented ones, morphable models have recently be-come very popular in Computer Graphics. They offer accessto high quality data through a practical parametrization thatbuilds on predictive parameters learned from statistical anal-

ysis. In [BV99], a morphable models of face 3D shapes andtexture is learned from hundreds of accurate laser scans ofhuman subjects. It offers control over intuitive parameterssuch as age, sex, mood, etc. Similarly a morphable modelof body shape has been proposed from laser scans of bodyshapes [ACP03]. Morphable models outperform simple scal-ing or FFD-like transformation by allowing to always main-tain the result within a plausible space characterized by thelearning examples.

For the first time, this paper investigates the generation ofa morphable model in the specific case of animation skele-tons. This raises the problem of using continuous interpo-lation over data that represents both rotational angles andlimbs lengths. In particular, the parameterization of 3D rota-tions may present singularities (such as gimbal lock for eu-ler angles) and is not unique (2π-periodicity), which makesits use more difficult in a statistical model. In addition, themorphable model has to take into account values definedin different units (e.g. distances and angles). Finally, to be

c© The Eurographics Association 2005.

L. Reveret & L. Favreau & C. Depraz & M.P. Cani / Morphable model of quadrupeds skeletons for animating 3D animals

Figure 1: Constructing learning data - the cow example

practical, a morphable model should offer appropriate con-trol parameters in the sense of being general enough to of-fer enough variability in the generated results and specificenough to maintain a good level of intuitive usability. Thispaper addresses these problems and shows how to build asuccessful and intuitive morphable model of skeletons forquadrupeds animals.

2. Related work

Jane Wilhems and Allen Van Gelder presented one of thefirst animation methods for animals [WG97]. Their approachis anatomical and thus relies on an accurate modeling ofbones, muscles and skin tissues. We target a different goal,namely the automatic construction of animation skeletons(defined as a hierarchy of joints) dedicated to the efficientanimation of animals through a standard geometry attach-ment method such as smooth skinning. We thus use a statis-tical analysis over a sample set of animation skeletons ratherthan anatomical modeling.

Wade and Parent use a medial axis computation to au-tomatically create geometric skeletons inside polygonalmeshes of animals [WP00]. The geometric skeleton iscleaned-up and automatically updated so that it can be usedfor character animation. Our approach is different since werather learn the placement of joints on reference skeletons,for which an expert animator has placed the joints in accor-dance with the anatomical data of the associated quadruped.As these examples show, most of the animator-defined jointsare not located onto the medial axis of the animal’s mesh (seefigure 1 for instance). Although less automatic, our approachguarantees that the skeleton we generate for a new animalwill be closer to the structure a skilled animator would havebuilt.

A model of animal growth has been proposed by Walteret al. [WF97, WFM01]. This approach covers the aspects ofthe skeleton, body shape and texture adaptation to representgrowth of the animal. We investigate here another source ofvariation, which is related to change in the skeleton mor-phology over different species of quadrupeds in our case.

Sumner and Popovic tackle the problem of retarget-

ing the animation of an animal towards another animal’smesh [SP04]. Motion warping is performed directly on tri-angle meshes, skipping the use of animation skeletons. Ourchoice is rather to focus on the underlying skeleton struc-ture as the basic component for character animation. Thisallows us to insert our morphable model into the standardwork-flow of 3D character animation.

The theoretical approach in Grochow et al.’swork [GMHP04] is closely related to ours. This workcombines a large set of human motion capture data in anelaborated probabilistic model which can generate posesfrom geometric parameters such as IK handles. In our case,although we target a statistical model controlled by similargeometric parameters, we work on morphological variationbetween skeletons of animals in rest poses rather than onmotion data. In consequence, we are not only trying tocapture the variation of joint angles, but also the positionsof these joints through the limbs length parameters. Theresulting mix of parameter units has raised specific problemsfor which we discuss solutions in this paper.

3. Learning data set

To be useful, morphable models need accurate data for thelearning phase. Laser scans and motion capture systems canprovide such data for human body and shape. Obtainingdata on animals skeletons is more challenging. As we tar-get ready-to-animate models, we have decided to learn themorphable model on reference skeletons built by an expertanimator (15 years experience in professional production).

3.1. Reference skeleton models

We have considered nine four footed animals covering abroad spectrum of morphologies : horse, goat, bear, lion,rat, elephant, cow, dog and pig. The reference skeletonswe are using have first been built from anatomical refer-ences [EDB56, Cal75]. The latter provide 2D drawings ofboth the animal’s internal and external anatomy, thanks tothe outline of the body shape and the structure of the anatom-ical skeleton for a rest pose. However, these 2D drawingsonly give side view information. In order to design full 3D



skeletons, 3D models of the animal’s shape have been usedas well. In order to ensure a correct alignment between dataduring the learning phase, all the skeletons share the sametopology in terms of number of articulations and joints hier-archy. We used the standard convention in animation of tak-ing the pelvis articulation as the root of the hierarchy. Eachskeleton consists in 58 articulations with 6 degrees of free-dom per articulation as they vary in position and orientationfor each subject. Figure 1 shows some steps of the skeletondesign in the example of a cow.

3.2. Parameterization of the data set

Each skeleton is parameterized as a single observation vec-tor containing position and orientation information for eacharticulation. One first concern with statistical analysis is toclearly specify which variance will be considered. The met-ric system of 3D animation is unit less: a mesh may haveany unit scale. We decided to get rid of any scaling effectover the data. Consequently, all the skeletons are normalizedso that pelvis articulations are at the same location, and thearticulation position are uniformly scaled so that the spinecolumn has the same length for every animal. This leavesvariability to be explored independently from the size of theanimal.

We have also considered two alternatives for features pa-rameterization:

• using euler angles versus quaternions for representing ro-tations angles

• using local axis coordinates versus global axis coordinates(i.e. world reference), for both translation and rotation val-ues

In the following sections, we show and discuss the impactof these choices. For clarity, we will refer to these conditionsas ER for Euler angles rotation, QR for Quaternion rotation,LC for local axis coordinate and GC for global axis coordi-nates.

Finally, each skeleton is represented by a 58 ∗ (3 + 3) =348 scalars vector for ER and 58∗(3+4) = 406 scalars vec-tor for QR. LC and GC are obtained by standard manipula-tion of transformation matrices, followed by matrix conver-sion into euler angles or into quaternion space. This leadsto 4 conditions to explore: ER×LC, QR×LC, ER×GC, andQR×GC.

When gathering data for all the animals, rotations parame-ters need a special care for dealing with discontinuities. Thestatistical analysis will linearly blend values of learning data,providing wrong results if two individual having similar ro-tation matrices are represented with very different parame-ters values, such as different 2π factors for Euler angles oropposite signs for quaternions. Moreover, Euler angles nearthe gimbal loose one degree of freedom leading to many-to-one problems. In each representation, ER or QR, these

problems are checked and corrected when the 9 examples ofthe learning set are gathered.

Finally, we stack all the data in a single X matrix. Wearrange the data in row vectors, each vector being a skeleton.X is thus a matrix of 9× 348 for ER (or 9× 406 for QR)scalars.

3.3. Normalization

Mixing data with different units such as rotation and trans-lation parameters raises the issue of normalization. Transla-tion data have been scaled so that the vertebral column hasa length of 10. Rotational data are expressed in radians forER condition and are left in the canonical range of [−1,+1]for the QR condition. The table below reports for each trans-lation and rotational parameters, the highest standard devia-tion computed over all animals chosen among all the articu-lations (for the sake of clarity we did not present the standarddeviation for each articulation). It gives an idea on a how thelearning data are scaled.

ER×LC ER×GC QR×LC QR×GC

tx 1.1 0.6 1.1 0.6ty 0.5 3.2 0.5 3.2tz 0.6 1.6 0.6 1.6rx 0.1 2.1 0.1 0.2ry 0.2 1.2 0.1 0.2rz 0.6 2.0 0.2 0.2rs n.a. n.a. 0.2 0.2

A common practice in statistical analysis consists in di-viding each observation parameters by its standard deviationacross the learning data set. Choosing a correct normaliza-tion can be stated as choosing the right trade-off between amodel where no translation data are taken into account (allanimals have the same limbs length) and a model where norotational data are taken into account (all animals have thesame articulations). In our case, it was not clear if one shouldbe favored with respect to the other one. We conducted sev-eral experiments on normalization values to be applied andwe did not report any significant impact on our ultimate goalof constructing a morphable model controlled by geometri-cal parameters.

4. Exploring dimensionality

Despite each skeleton is parameterized by 348 (or 406)scalars, the data set of 9 examples gives the model a max-imum number of 8 linear degrees of freedom plus a meanpose. What we are interested in this section is to give asense of redundancy between examples. The question iswhether this redundancy can be factorized in less controlparameters than 8 (similarly to shapes blending, skeletonsblending would be a canonical parametrization of a mor-phable model). We studied this problem by first applying a



1 2 3 4 5 6 7 80

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sum=93.7%

sum=96.5%

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horscowliondogbeargoateleratpig

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Figure 2: Principal components distribution in 4 parameterizations

standard PCA to the data set for each of the 4 conditions{ER,QR}×{LC,GC}. In figure 2, we show the distributionof the principal components over all the subjects in the 4features parameterization conditions. Let us detail how thisdistribution has been estimated.

Usually, PCA results are discussed in terms of the cumu-lative amount of variance of the data, by adding successiveprincipal component. Figure 2 follows this idea but adds de-tails to analyze results more precisely per individual of thedata set. What is represented is computed as follow. PCAconsists in computing the p eigenvalues D and p eigenvec-tors E of a data set X of n elements. X is centered on themean skeleton vector. In our case, we have a maximum ofp = 8 eigenvectors related to non-zero variance. E stackseigenvectors in column.

1n

XtXE = ED (1)

EtE = 1 (2)

D is a p× p diagonal matrix where each element gives thetotal variance of the data explained by each of the p princi-pal components. The total variance of the data set is given

by trace(D). Each column of the matrix P = XE gives theprojection of all individuals on each eigenvector. Matrix Dcan be interpreted as:

D =1n

PtP (3)

If we expand D along the row vectors Pi· of P, we get:

D =1n

n

∑i=1

Pti·Pi· (4)

Finally, as D is diagonal, we have for each element D j:

D j =1n

n

∑i=1

P2i j (5)

The variance of each component can thus be decomposedaccording to the contribution of each one of the n individu-als. On figure 2, we plot the squared value of the projectionP2

i j of each individual in function of the j-th component,normalized by the total variance trace(D). In addition, weindicate with a vertical line, at which component the cumu-lative all-individuals variance explained by successive com-ponents reached 90% and 95% of the total variance.



What we learn with figure 2 is that some parameterizationleads to some over-fitting cases. Indeed, especially in the LCcases, the variance of a given component may be mostly re-lated to a single individual. It means that this component isnot factorizing information across the data set. The best re-sults are obtained for the GC×QR condition (world coordi-nates and quaternion parameterization). In this parameteri-zation, few over-fittings occur. Furthermore, 90% of the to-tal variance is already captured using the 3 first components.This result suggests that, under the GC×QR condition, allthe data can be efficiently linearly packed into a linear modelcontrolled by only 3 parameters.

A new animation skeleton is simply generated by linearcombination of the eigenvectors added to the mean skeletonX of the learning data set:

x(p) = X+pEt (6)

On figure 3, we show the results of the variation of themorphable model along the three first linear modes at minustwo/one and plus two/one times the standard deviation ofthe parameter. The central column corresponds to the meanshape.

Figure 3: Three first modes of variation of PCA on skele-tons. Surprisingly, these modes are easy to identify as ani-mal height in rest pose, bending of vertebral column and ahoofed vs plantigrad parameter.

Although these parameters can be implicitly interpreted asanimal height, bending of vertebral column and a hoofed vsplantigrad parameter, they would not be easy to align on a 3Dmodel: their optimal values would be difficult to guess man-ually for a given mesh. We conclude this section by keepingin mind that 3 parameters might be sufficient. The next sec-tion investigates 3 parameters with a more practical usabilitywhere the morphable model is controlled explicitly by geo-metrical parameters.

5. Geometrical parameterization

In [ACP03], Allen et al. mention that even if PCA providescompact parameters, they are not always intuitive to use. In-

stead, their space of 3D body shapes is controlled with pa-rameters such as weight and height to provide a more usablemorphable model. We share the same approach for our mor-phable model of quadruped skeletons, making the alignmentof the skeleton on a an arbitrary mesh much easier to per-form. In our case, the control parameters are intuitive geo-metric values that can be measured on an animal’s anatomi-cal skeleton in side view (see figure 4). Indeed, similar geo-metric measurement can be pointed on the 3D mesh and ap-plied to the morphable model. The generated skeleton willthen be aligned with the mesh, granting that the morphablemodel is stable - in the sense that the geometric control pa-rameters should generate a skeleton having the same geo-metrical measurement.

Figure 4: The three geometrical measurement controllingthe morphable model

Based on the observation of the three main PCA modes,three geometrical parameters have been tested (to be com-pared with the 9 parameters used in [WF97, WFM01] foranimal growth):

• animal height m1, measured on a mesh as the vertical dis-tance between an estimated location of the pelvis and thefloor;

• vertebral column bending m2 (we remind that vertebralcolumn length is kept constant and serves as a scaling fac-tor) measured as the difference between the animal height(defined as above) and the height of base of its neck; forthe sake of clarity, the video shows an arrow from theground to the neck for this parameter (similarly on thecolor plate). The true measurement is the one firstly de-scribed in this paragraph.

• hoofed vs plantigrad parameter, measured by the angle m3between floor plane and the line joining the rear foot to therear ankle.

These parameters are extracted on the reference skeletonsby computing the global transformation matrices on pelvis,first neck cervical and rear ankle. As a first model, we per-form a simple linear mapping from these three measure-ment parameters to skeletons rotation and translation. Moreelaborated mapping could be applied, such as Radial Basis



Functions (RBF) [LCF00] or Scaled Gaussian Process La-tent Variable Model(SGPLVM) [GMHP04]. Linear mappingproved to be sufficient in our case. The mapping is done froma 3 scalars measurement vector m = [m1,m2,m3] to a skele-ton vector x, both arranged as row vectors:

x(m) = X̄+mV (7)

The linear model V is estimated by a least squares fittingbetween reference skeletons X and the matrix M of the threeparameters measured on the reference skeletons stacked asrow vectors. This leads to:

V = (MtM)−1MtX (8)

Figure ?? shows the variations of this morphable modelalong its three control parameters. Note that contrary to thethird PCA mode, the third parameter correlates hoofed vsplantigrad with the length of neck (this correlation is indeedexisting on the animal examples we provided; we did notinvestigate yet whether it is valid on all quadruped, whichcould be true for anatomical reasons). The neck length isitself correlated with the bending of the vertical column, asshown by both the second PCA mode in figure 3 and thesecond geometric parameter in figure 5.

Figure 5: Three first modes of variation of the morphablemodel controlled by animal height, vertebral column bend-ing and hoofed vs plantigrad parameters (foot angle).

Any parameters could be used as control parameters of themorphable model. We made some experiments in which theneck length was measured instead of the hoofed vs planti-grad parameter. It results in keeping the neck length con-stant when modifying the bending of the vertebral columnas shown on figure 6.

Figure 6: Three first modes of variation of the morphablemodel controlled by animal height, vertebral column bend-ing and neck length.

6. Results and discussion

6.1. Reconstruction of the data-base

We discuss in this section the impact of the 4 conditionsER,QR× LC,GC on the properties of the morphable mod-els obtained from both PCA and from our geometrically-controlled model. We evaluate the results based on howwell the learning examples are correctly reconstructed bythe morphable model from their associated input parame-ters. In the case of the PCA model, these inputs parametersare the projection coefficients of the learning examples ontothe eigenvectors. In the case of the morphable controlled bygeometrical parameters, these input parameters are the mea-surements made on the skeletons. As the size of the learningset is small and the number of parameters (three) is inferiorto the number of learning examples (nine), non exact match-ing can be expected in both cases. We separate the evaluationin two parts:

1. error on translations2. error on rotations

For the error on translations and rotations, all skeletonsdata, both reference and predicted using one of the 4 condi-tions, are re-encoded into a common representation to com-pare the 4 conditions on the same basis: global 4× 4 trans-formation matrices are computed and split into their transla-tion part (fourth column 3×1 vector) and rotation part (first3×3 sub-matrix). For each individual, the error is estimatedas the maximum over all the articulations of the norm of thedifference between the reference skeleton and the predictedskeleton, computed on the translation vectors for translationand on rotation matrices for rotation (largest eigenvalue ofthe SVD is used in this case). Figure 7 (respectively Fig-ure 8) summarizes the results for all the 9 examples underthe 4 conditions using the PCA model controlled by the firstthree principal components (respectively the geometrically-



controlled model). They show that the reconstruction of thedata set is as good with our geometrically-controlled modelthan directly using PCA.

ER,LC QR,LC ER,GC QR,GC0.5

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Figure 7: Mean and standard deviation of the prediction er-ror for the 4 conditions: translation (left), rotation (right)using the PCA-controlled model


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Figure 8: Mean and standard deviation of the prediction er-ror for the 4 conditions: translation (left), rotation (right)using the geometrically-controlled model

For each of the models, these results show that the erroron translation is smaller with the use of a global frame (GCcondition), whatever the representation of rotations. Thiscomes from the fact that in LC conditions, prediction er-rors are cumulative due to the hierarchical structure of theskeleton. This has maximum effect at limbs ends. The GCconsition induces larger variation on rotational data. In thiscase, quaternions offer more stability than Euler angles. Thisexplain the better results of GC×QR compared to GC×ER.Figure 9 illustrates these two representations of data in theworst prediction case, which occurs for the elephant. To con-clude, these observations suggests that global transforma-tion parameters and quaternions representation are the bestchoice to build a morphable model of animals skeletons.

6.2. Animation of a new quadruped

In order to validate the whole process – from the skeletongeneration to the resulting animation – we applied our modelto an animal that was not present in the learning database:a cat. We generated the skeleton using the geometrically-controlled model from measurements taken on a side viewof the cat mesh. The cat skin was then rigged to the skeletonusing smooth skinning. An example of a final animation isprovided in the video. In addition, the video illustrates the

Figure 9: Comparison between conditions QR × LC (left)and QR×GC (right) in the elephant case. The reference isdepicted in purple and the predicted skeleton in blue.

capabilities of motion retargeting offered by the morphablemodel. The same walking animation is applied on the skele-ton while the parameters of the morphable model are contin-uously edited during the sequence.

7. Conclusion

This paper has shown that statistical analysis can success-fully be used for the automatically generation animationskeletons. We have tackled the problem in the specific caseof four footed animals, using some skeletons built by an ex-pert animator as the learning database as well as for valida-tion. Our results have shown that a three dimensional spaceis sufficient for representing this set of animation skeletons.The resulting morphable skeleton model can easily be fittedto any quadruped by taking three simple measurements on aside view on an animal mesh. We have shown that it yieldsconvincing animation results when combined with standardsmooth skinning for geometry attachment. Indeed, expertanimators may use this automatically generated skeleton asa starting point and probably reach a higher level of realismthrough fine tuning dedicated to each animal.

References

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[BV99] BLANZ V., VETTER T.: A morphable model forthe synthesis of 3d faces. In SIGGRAPH ’99: Proceedingsof the 26th annual conference on Computer graphics andinteractive techniques (New York, NY, USA, 1999), ACMPress/Addison-Wesley Publishing Co., pp. 187–194.

[Cal75] CALDERON W. F.: Animal painting and anatomy.Dover Publications, Inc., New-York, 1975.

[EDB56] ELLENBERG W., DITTRICH H., BAUM H.: Anatlas of animal anatomy for artists. Lewis S. Brown (eds),Dover Publications, Inc., New-York, 1956.



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[SP04] SUMNER R. W., POPOVIC J.: Deformation trans-fer for triangle meshes. ACM Transactions on Graphics(SIGGRAPH’04) 23, 3 (August 2004).

[WF97] WALTER M., FOURNIER A.: Growing and ani-mating polygonal models of animals. Computer GraphicsForum (EUROGRAPHICS’97) 16, 3 (1997).

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[WG97] WILHELMS J., GELDER A. V.: Anatomicallybased modeling. In Computer Graphics (ACM SiggraphProceedings) (1997).

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